Iterating 5-6/x and graphing the limits of
these
sequences
2b.We'll iterate the function
starting with different numbers, then graph
the limits of the infinite sequences we get.
Starting number 1. Putting 1->x, we get 5-(6/1)=-1. Putting this output
number -1 in for x now, we get 5-(6/-1) = 11. Putting 11 -> x we get
5-(6/11)= 4.45..
So we get this infinite sequence:
1, -1, 11,
4.45.., 3.653..., 3.3575..., 3.2129..., ...which goes to 3 as the
limit.
Starting no. -100, 5.06, 3.81422..., 3.4269...,
3.2491...,3.1533..., ...which goes to 3 as the limit.
Notice that starting with 2 we get a constant sequence 2, 2, 2, ... On
the graph 2 goes to 2.
All other starting numbers lead to an infinite
sequence which goes to 3 as the limit.., except an infinite number of
starting numbers like zero, and 6/5 that at some point make the denominator
go to
0 and make the fraction 'blow up' and thus has no answer. These numbers
put a hole in the graph.Can you find a rule for these numbers that
make the denominato 0? What is the limit of the sequence of these numbers?
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